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Fig. 9.2
Neighborhoods in
the (
Auber et al.
,
2003
)
strength clustering method
W
uv
are the common neighbors of
u
and
v
,
M
u
=
N
u
\
N
v
are neighbors of
u
but not
neighbors of
v
,and
M
v
=
N
u
are neighbors of
v
but not neighbors of
u
(Fig.
9.2
).
For this operation, a ratio is calculated based on the number of cycles of length 3
involving an edge
e
:
N
v
\
|
W
uv
|
(
)=
γ
e
3
|
M
u
|
+
|
W
uv
|
+
|
M
v
|
A similar ratio can be defined for cycles of length 4:
γ
(
e
)=
s
(
M
u
,
W
uv
)+
s
(
M
v
,
W
uv
)+
s
(
M
u
,
M
v
)+
s
(
W
uv
,
W
uv
)
4
Σ
(
)=
γ
(
)+
γ
(
)
.
Once again, the operation is a divisive method that maximizes
Q
. A quality
measure is used to neglect edges with low strength values (Fig.
9.3
).
This metric completes Burt's network constraint, which measures how much an
actor is constrained by its direct neighborhood (
Burt
,
2000
,
2005
). In both cases,
edges with low values are expected to act as bridges between tighter communities.
The strength index is then the sum of both indices
e
e
e
3
4
9.4
Comparison of Hierarchical Network Analyses
The three methods were compared by constructing datasets to describe the flow of
air transport passengers between cities, based on data from ENAC (French National
School of civil aviation). This data source adds national and airport sources to the
classical OAG source (Official Airline Guide) for completeness. We also aggregated
airports by city to reconstruct the actual traffic of each city.
Clustering using the Weighted Quality Measure for the international air-traffic
network, whether non-valued (
Guimerà et al.
,
2005
) or valued, demonstrates that
groups of cities are geographically organized by continents (Fig.
9.4
).
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